Method for determining the registration of a structure on a photomask and apparatus to perform the method

ABSTRACT

A method for determining a registration error of a feature on a mask, including providing a first aerial image that was captured by means of a position measuring device and includes at least the feature, simulating, from pattern specifications of the mask, a second aerial image that includes at least the feature, taking into account at least one effect that causes distortion of the first aerial image, and determining the registration error of the feature as the distance of the position of the feature in the first aerial image from the position of the feature in the second aerial image. Also provided is a method for simulating an aerial image from pattern specifications of a mask and a position measuring device for carrying out the method.

CROSS REFERENCE TO RELATED APPLICATIONS

Pursuant to 35 U.S.C. §119, this application claims the benefit of U.S.provisional application 61/381,700, filed Sep. 10, 2010 and Germanapplication DE 102010045135.5, filed Sep. 10, 2010. The aboveapplications are incorporated by reference.

BACKGROUND OF THE INVENTION

This patent specification is directed to a method for determining aregistration error of a feature on a mask.

This patent specification is also directed to a method for simulating anaerial image from pattern specifications of a mask.

The patent specification is also directed to a position measuring devicefor carrying out the method.

In lithography for semiconductor component fabrication, the patterns ofmasks, which are also referred to synonymously as reticles, areprojected by means of scanners or steppers onto wafers coated with alight-sensitive layer, the resist. Masks can be configured, for example,as “binary masks,” with chromium patterns on quartz glass, or asphase-shift masks. For EUV lithography applications, reflective masksare used. Templates for the nanoimprint process are another variety ofmask. With the aid of mask inspection microscopes or position measuringdevices, the pattern of a reticle is projected by means of lenses onto alight-sensitive local-resolution detector, such as, for example, a CCD(Charge Coupled Device) chip.

By means of a position measuring device (registration tool), features ona mask that are specifically designated as registration patterns or asmarkers, for example squares, crosses or angles of specific shapes, aremeasured and compared with their nominal positions. Positions on themask of features that are details of the patterns used on the mask arealso measured. This is known as “real pattern registration.” Thedeviation of the nominal position of a feature from its actual positionon the mask is the placement error, which is also known as the“registration” or “registration error.”

The measurement of masks makes it possible to check the positionalaccuracy of the patterns on the mask during the electron-beammask-writing process. In addition, by measuring the patterns of anexisting set of masks, the deviations of the pattern positions on thevarious masks can be adjusted to each other for the individuallithographic layers.

To check the positions of features, an aerial image of a detail of amask is captured with a position measuring device. The mask rests forthis purpose on a stage (also known as an object table or traversingunit) that serves to move the mask in the direction of the mask plane,in order to position a desired detail in the image field of thepositioning measuring device so that the image can be captured with adetector. Before the measurement is performed, the mask is oriented onthe stage in such a way that its position on the stage is known.Alternatively, relative alignment with special alignment features can beperformed. The position determination is then made in relation to thesefeatures, which are also known as alignment markers. The image can thusbe assigned unequivocally to the absolute or relative position of thedetail on the mask. By determining the position of the pattern in thecaptured image, it becomes possible to compare the nominal and actualpositions of the patterns on the mask, and thus to calculate theplacement error.

Metrological requirements for placement error determination currentlystand at 1 nm. Feature placement errors can be determined, for example,via methods for determining the positions of these features in aerialimages. Examples of the known methods are threshold value methods andcorrelation methods. Accuracy requirements for the to-be-determinedposition or placement error of a feature are higher than the resolutionof an aerial image captured by the position measuring device. In currentdevices, the edge lengths of a pixel of an aerial image correspond, forexample, to 25 nm in mask dimensions. Methods have been developed thatpermit so-called subpixel-accurate position determination of a featurein an aerial image.

For example, DE1 2006059431 describes a method for determining theposition of a pattern on a substrate with subpixel accuracy in relationto a reference point on the substrate by overlaying the pattern with areference pattern.

Known from DE10047211 is a method for determining the positions of edgesof a pattern with subpixel accuracy. Here, the measured edge profilesare compared with model intensity profiles to find the position of bestfit.

DE10337767 discloses a method for determining the distance between areference element and a pattern for measuring the overlay shift.

DE102007033815 discloses a method for determining the distance between areference element and a pattern by means of two pattern elements formeasuring the overlay shift, wherein in the reference element or in thepattern, one of the pattern elements whose position is to be determinedis amplified.

These methods have not yet made it possible to achieve the requirementsimposed on the next generation of position measuring devices.

SUMMARY

In one aspect, the invention provides an efficient method that permitsaccurate determination of the registration error of a feature on a mask.For example, metrological requirements for registration errordetermination can be as small as 0.5 nm. Specialized aerial imageanalysis methods are used to satisfy these requirements.

In another aspect of the invention, a method is provided for accuratedetermination of a registration error of a feature on a mask. The methodcomprises the steps of:

-   -   providing a first aerial image that was captured by means of a        position measuring device and includes at least the feature,    -   simulating, from pattern specifications of the mask, a second        aerial image that includes at least the feature, taking into        account at least one effect that causes distortion of the first        aerial image, and    -   determining the registration error of the feature as the        distance of the position of the feature in the first aerial        image from the position of the feature in the second aerial        image.

This measure has the advantage that registration errors introduced asartifacts by distortions occurring during the generation of the aerialimage are taken into account in the analysis of the first aerial image.Registration error results thus are not falsified by these distortions.

The aerial image is, for example, captured with a two-dimensionaldetector and provided as a two-dimensional matrix composed of pixels.Intensity values of the individual pixels are represented by numericalvalues.

Features on the mask can be, for example, crosses or squares, which canbe written onto the mask for inspection purposes. The method is alsosuitable for determining the registration errors of device patterns ordetails thereof, for example an edge or a corner. The method accordingto the invention can also be used to measure line widths. The linewidth, and thus its deviation from a nominal value, is derived from thepositions or registration errors of the two edges of a line.

The first aerial image includes a detail of the mask. This can bedefined by means of an image field of the position measuring device. Theregion of the second aerial image can be identical. It can also bedifferent, for example smaller than the detail of the first aerialimage, for example limited to the feature.

The position of the feature in the first aerial image represents theactual position of the feature on the mask, falsified by distortion ofthe aerial image. The position of the feature in the second aerial imagerepresents the nominal position of the feature on the mask, falsified bydistortion of the aerial image. The distance between the positions ofthe first and second aerial images is no longer affected by distortion.

The method can also include the capture of the first aerial image.

The second aerial image is determined by simulation. All effects thatcan occur in the optical beam path of the position measuring device,including the mask, can be taken into account here, in whole or in part.These effects include, for example, characteristics of the illumination,such as the wavelength, the intensity, the polarization or the degree ofcoherence of the illuminating light, the illumination setting used, thefield homogeneity of the illumination intensity. Characteristics of themask, such as, for example, the three-dimensional topography (the heightrelief) of the mask or the complex refractive index of the maskmaterial, are also included in the simulation. Examples of the effectsof the imaging beam path include: the numerical aperture and the imagingerror of the objective lens, pupil apodization, and polarization.Proximity effects are also included in the simulation.

In a further embodiment of the invention, at least one of the followingeffects is taken into account in the simulation of the second aerialimage: proximity effects, distortion due to an objective lens of theposition measuring device, topography of the mask, and refractive indexof the mask.

This measure has the advantage that essential effects are taken intoaccount, but computing complexity is still limited. It is feasible totake these effects into account in the simulation of aerial images.

Proximity effects may cause considerable distortion, especially whenmeasurements are being performed on small patterns close to the limit ofresolution of a position measuring device.

Proximity effects are to be described as resolution-induced intrinsicdistortions. These are attributable to the nature of thediffraction-limited resolution of a projected image. The causes lie notonly in the environment of a pattern, but also in the pattern itself.These effects can be taken into account in the simulation.

Distortion due to the objective lens of the position measuring devicecan be determined and therefore readily taken into account in thesimulation of the second aerial image.

The topography of the mask and the refractive index or complexrefractive index of the mask material can also be determined and takeninto account in the simulation.

In a further embodiment of the invention, the simulation of the secondaerial image is based on a mask pattern that has, at least in part, ahigher resolution than the first aerial image.

This measure has the advantage the discretization errors in thecalculation of the aerial image are minimized.

Since the positions of features in the first and second aerial imagesare to be determined with subpixel accuracy, a method that produces aposition-sensitive aerial image is advantageous for simulating thesecond aerial image. The mask pattern on which the aerial imagesimulation is based is in the form of a transmission function or areflection function. Hereinafter, the term “transmission function” alsoencompasses the term “reflection function.”

To minimize computing complexity, in a variant of this embodiment, thecalculation of the aerial images using this high resolution of thetransmission function of the mask is limited to regions relevant to theposition determination. The resolution of the mask pattern, i.e., thetransmission function of the mask, is preferably 1 nm/pixel or 0.5nm/pixel. The stated measurements refer to the dimensions of the mask.The pixel size refers to the resolution of the first aerial image and isdefined by the design of the detector of the position measuring device.

In a further embodiment of the invention, the simulation of the secondaerial image is based on pattern specifications of the mask that are inthe form of a continuous transmission function, the second aerial imagebeing determined as a matrix of discrete intensity values, adiscretization of the spectrum of the mask being performed withoutfurther approximation.

This measure has the advantage of providing an efficient method for thesimulation of position-sensitive aerial images. Displacements offeatures due to errors caused by a discretization of the transmissionfunction are avoided.

In an advantageous variant of this measure, the continuous transmissionfunction is decomposed into a sum of functions. The individual functionsdescribe basic building blocks of the pattern specifications ortransmission functions of the mask. The building blocks are configured,for example, as triangles, since all polyhedra deriving from the patternspecifications of current masks can be decomposed into triangles.

In a further embodiment of the invention, at least one region of thefeature is defined that will be taken into account in determining thedistance.

This measure has the advantage that regions that are particularlyposition-sensitive are selected. This increases the accuracy of themethod while at the same time minimizing computing complexity.

Plural regions of the feature can also be selected. This increases theaccuracy of the pattern determination.

In a further embodiment of the invention, the at least one region of thefeature is embodied as an edge of the feature.

This measure has the advantage that particularly position-sensitiveregions can be included in the position determination. Due to thecomparatively high intensity gradients of edges of features in aerialimages, these change radically with position.

In a further embodiment of this measure, plural edges are selected tofurther increase the accuracy of measurement.

In a further embodiment of the invention, the method includes the stepsof:

-   -   designating, in the first and the second aerial image, at least        one region of the feature that is to be taken into account in        determining the registration error,    -   determining the at least one first and second position of the at        least one region, and    -   determining the distance on the basis of the at least one first        and second positions of the at least one region in the aerial        images.

This measure has the advantage that the same method for determining thepositions of the regions (the first and the second position) can be usedon both aerial images (the first aerial image and the second aerialimage). In this way, there are no discrepancies between differentmethods that would falsify the registration errors.

In a further embodiment of the invention, the position of the feature isdetermined by means of the position of at least one edge of the feature,the position of said edge being determined by a threshold value method.

This measure has the advantage that the position of the feature can bedetermined in a simple and fast manner.

The threshold value method, also known as the threshold method, makes itpossible to determine the position of an edge without having to know theprofile of an edge. The position of the edge is determined from an edgeprofile using a threshold value of, for example, 70% of the maximumintensities in the aerial image.

The above method can be used to directly determine the position of thedesignated region in an aerial image for a defined threshold value.Subpixel-accurate position determination is also possible.

In a further embodiment of the invention, the method comprises the stepsof:

-   -   designating at least one region in the first or the second        aerial image,    -   determining a position of the region in the other aerial image,        and    -   determining the distance on the basis of the at least one        difference between the positions of the at least one region in        the aerial images.

This measure has the advantage of providing high-accuracy methods fordetermining the respective positions of regions of an image in anotherimage.

The at least one region can be designated in the first aerial image andthe position of the region determined in the second aerial image. Theregion can also be designated in the second aerial image and theposition of the region determined in the first aerial image.

The at least one difference between the distances separating thepositions for a feature and the pattern specifications of the mask canbe used to determine the actual position of this region.

The positions can be determined with subpixel accuracy by this method.

In a further embodiment of the invention, the position of the at leastone region is determined by the least squares method.

This measure has the advantage of making it possible also to determinethe positions of regions that have intensity discontinuities in theiredge regions.

These discontinuities make it impossible to use cross-correlation forposition determination, since the Fourier spectrum of such images is notbandwidth-limited. This measure makes it possible to determine positionsof arbitrary details of patterns, including details of device patterns,such as corners or edges, for example, in aerial images.

In a further embodiment of the invention, the at least one designatedregion is masked, all the image data of the aerial image beingsuppressed outside the masked region.

This measure has the advantage that a simple and fast method can be usedfor position determination. Since the size of the first and secondaerial images remains the same because of the masking, the use ofcorrelation methods is greatly simplified.

Other regions in an aerial image can also be masked.

In a further embodiment of the invention, a first position of the atleast one edge derived from the pattern specifications of the mask isadopted as the nominal position. A second position of the edge isdetermined as the actual position, based on the pattern specificationsof the mask and the difference between the positions of the at least oneregion in the aerial images.

This measure has the advantage that the nominal and actual positions ofedges can also be determined on the basis of correlation methods.

From the positions of the edges the position of a feature can also bedetermined, for example as the position of the center of symmetry.

In a further embodiment of the invention, the method comprises the stepsof:

-   -   determining a first position of the feature in the first aerial        image, from at least one first position of an edge,    -   determining a second position of the feature in the second        aerial image, from the at least one second position of an edge,        and    -   determining the distance as the difference between the first        position and the second position.

This measure has the advantage of providing an efficient method foridentifying features from edge positions.

The position of a feature can be determined, for example, from thepositions of the center of symmetry. A characteristic position of thefeature can also be determined by statistical methods, such as the leastsquares method, for example, using the positions of plural edges.

In a further embodiment of the invention, the at least one regionincludes the feature.

This measure has the advantage that only one position determination needbe performed. Designating the detail simultaneously serves the purposeof designating the feature whose registration error is to be determined.

Further according to the invention, a method is provided for simulatingan aerial image from pattern specifications of a mask, wherein theaerial image is determined as a matrix composed of pixels havingintensity values, taking as a basis the pattern specifications of themask in the form of a transmission function that has, at least in part,a higher resolution than the first aerial image.

This method makes it possible to simulate position-sensitive aerialimages. Numerous advantages of the method and of the followingembodiments have already been stated.

In a further embodiment of the invention, the pattern specifications ofthe mask are present in the form of a continuous transmission function,and a discretization of the spectrum of the mask is performed withoutfurther approximation.

In a further embodiment of the invention, the continuous transmissionfunction is decomposed into a sum of functions that describe the basicbuilding blocks of the pattern specifications.

In a further embodiment of the invention, the building blocks areconfigured as triangles.

In a further embodiment of the invention, the pattern specifications ofthe mask are transformed into a matrix.

This measure has the advantage that the aerial image can be simulated ina simple manner by determining the complex spectrum of this matrix.

In a further embodiment of the invention, the resolution matrix varies.

This measure has the advantage that the computing complexity forsimulating the aerial image is limited. The resolution of the matrix canbe increased in regions with higher intensity gradients (such as, forexample, in the vicinity of edges).

Further according to the invention, a position measuring device isprovided, comprising a control unit that performs the following steps:

-   -   providing a first aerial image that is captured by means of the        position measuring device and includes at least the feature,    -   simulating a second aerial image that includes at least the        feature on the basis of pattern specifications of the mask,        taking into account at least one effect that causes distortion        of the first aerial image, and    -   determining the registration error of the feature as the        distance of the position of the feature in the first aerial        image from the position of the feature in the second aerial        image.

The control unit is, for example, a computer programmed such that theaforesaid steps can be performed. The control unit programmed such thatall the aforesaid methods and their embodiments can be carried out.

For example, the illumination setting and the polarization of theilluminating light from the position measuring device are preset for thecapture of the first aerial image. These presets can be taken intoaccount in the simulation of the second aerial image.

It is understood that the characteristics of the invention cited aboveand explained in more below can be used not only in the describedcombinations, but also in other combinations, without departing from thescope of the present invention.

The invention will be described and explained in more detail below onthe basis of a few selected exemplary embodiments and with reference tothe drawings.

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a position measuring device.

FIG. 2 is an aerial image of a feature.

FIG. 3 is a graph showing the discrete intensity profile of the aerialimage along section line A-A in FIG. 2.

FIGS. 4-8 are graphs showing representations explainingsubpixel-accurate position determination.

FIGS. 9-11 are graphs showing representations explaining anotherembodiment for subpixel-accurate position determination.

FIGS. 12-14 are graphs showing representations explaining anotherembodiment for subpixel-accurate position determination.

FIG. 15a shows a mask pattern of a first example for determiningregistration errors.

FIG. 15b shows an aerial image of the first example for determiningregistration errors.

FIG. 16 is a graph of registration errors introduced as artifacts byproximity effects in the first example.

FIG. 17 shows a mask pattern of a second example for determiningregistration errors.

FIGS. 18a and 18b are aerial images of the second example fordetermining registration errors.

FIG. 19 are graphs of registration errors introduced as artifacts byproximity effects in the second example.

FIG. 20 is a flow chart of an exemplary embodiment of the methodaccording to the invention.

FIG. 21 shows an example of the decomposition of a transmission functionof a mask into triangular regions.

DETAILED DESCRIPTION

FIG. 1 depicts a position measuring device 10 that serves to measure theposition of patterns on masks.

A mask 1 for photolithography is placed on a stage 2. The stage 2 can bemoved in three spatial directions to position the mask 1. To ensure highaccuracy, the current position or the path difference is monitored bymeans of laser interferometric or other high-precision measuringinstruments (not shown). The mask 1 and the stage 2 are arrangedhorizontally; the mask plane is denoted the xy-plane. Disposed above thestage 2 with the mask 1 is an illumination device 3. This includes atleast one illumination source emitting coherent illuminating light,which illuminates the mask via an illumination beam path. Theilluminating light source can be embodied, for example, as a laser thatemits light with a wavelength of 193 nm. The illumination device 3serves to shine light through the mask 1. On the other side of the stage2 is another illumination device 3′, which serves to illuminate the mask1 with incident light. Stops 11 and 11′ are used to configureillumination settings, such as, for example, dipole, quadrupole,annular, etc. The polarization of the illuminating light is set viapolarizers 12, 12′. To center the stop 11 or 11′, a Bertrand lens 13 isbrought into the beam path by drive 14. The pupil plane of the imagingbeam path of the position measuring device 10 is then projected onto adetector 6.

A detail of the mask 1 that is located in the image field is projectedthrough a projection lens 4 and a beam splitter 5, using either thelight passing through the mask 1 or the light reflected by it, onto thelocal-resolution detector 6 embodied as a CCD (charge coupled device)camera. The optical axis of the projection lens 4 is denoted byreference numeral 9, and its direction is designated the z-direction.The detected intensities of the first aerial image are digitized andstored as a grayscale image by a control unit 7, embodied as a computerwith a monitor. This image is configured as a matrix of 1000×1000 pixelsbased on intensity values.

Measurement of the positions of patterns on the mask 1 is performed inrelation to reference markings (alignment markers) on the mask 1. Thepositions of patterns on the mask are stated in terms of the maskcoordinate system. This is defined by two mutually perpendicular axesdenoted x and y. To measure the positions of the patterns relative tothe mask surface, before the individual positions of the patterns aremeasured, the position of the mask on the stage is determined. This canbe done, for example, by determining the respective positions of theedges of the mask 1 in relation to special features, known as markers,whose positions on the stage are known. The laboratory coordinate systemis defined by the positions of the markers on the stage. The coordinatesystem of the image field of the first aerial image to be captured andof the first aerial image itself is a right-angled coordinate systemwhose x- and y-axes are parallel to the axes of the mask coordinatesystem. The origin is at the center of the image. The position of theaerial image or of the image field in relation to the mask is given bythe position of the origin in the mask coordinate system.

The following discussion relates to the mask coordinate system. Asstated above, it is possible to convert the positions to other referencesystems, such as, for example, the laboratory coordinate system.

To take an aerial image of the mask 1, the latter is oriented in thexy-plane so that the desired region comes to lie in the image field ofthe position measuring device and is projected onto the detector 6. Oncethe best focus plane has been determined by moving the stage 2 in thez-direction, a grayscale image is taken by the detector 6 and thecontrol unit 7. In a variant of the method, additional aerial images aretaken parallel to the best focus plane; these are known as focus stacks.For example, 7 or 9 aerial images are taken at distances from the bestfocus plane in both directions of the optical axis, for example atintervals of 150 nm. The registration error determination is performedindividually for each of the images in a focus stack. The imagesharpness is also evaluated for each focus stack. The registration erroris used to generate a function that depends on image sharpness. Thevalue of this function at the point of best image sharpness is the mostaccurate value of the registration error.

The simulation of the second aerial image is performed by means ofcommercially available software, such as MicroSim, for example, based onthe pattern specifications of the mask, the mask design. The MicroSimsoftware is described, for example, in: M. Totzeck, “Numericalsimulation of high-NA quantitative polarization microscopy andcorresponding near-fields,” Optik 112 (2001), 381-390(MicroSim-Software, University of Stuttgart). The imaging conditions ofthe position measuring device, such as, for example, numerical aperture,wavelength, degree of coherence of the illumination, etc., are takeninto account in the simulation. Effects that cause distortion of theaerial image during the imaging of the mask by the position measuringdevice, particularly proximity effects, are taken into account. Withregard to proximity effects and aerial image simulation, see thearticle: H. H. Hopkins, “On the diffraction theory of optical images,”Proceedings of the Royal Society of London, Series A, Mathematical andPhysical Sciences 217/1130 (1953), 408-432.

To carry out the method, in a variant, a simulation of the second aerialimage with higher resolution of the transmission function of the mask isused. This measure particularly has the effect of minimizingdiscretization errors in the calculation of the aerial image. The extentof a pixel of the detector 6 in the x- and y-directions is consistently25 nm, referred to the dimensions of the mask. Calculation of atransmission function with the resolution at the ideally needed 1nm/pixel results in a matrix of 25,000×25,000 pixels. To minimizecomputing complexity, only the relevant regions of the aerial image arecalculated using this high resolution of the transmission function orreflection function of the mask.

The procedure used to simulate the aerial images is as follows. Theintensities of an aerial image composed of a matrix of n×m pixels aredetermined for each illumination direction j as:

$\begin{matrix}{I_{nm}^{(j)} = {{\sum\limits_{n^{\prime} = 1}^{N}{\sum\limits_{m^{\prime} = 1}^{M}{S_{n^{\prime}m^{\prime}}P_{n^{\prime}m^{\prime}}^{(j)}{\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({n^{\prime} - 1})}}{{({n - 1})}/N}}{\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({m^{\prime} - 1})}}{{({m - 1})}/M}}}}}}^{2}} & I\end{matrix}$The total intensity is obtained as the weighted sum over all theillumination directions j, where ω_(j) are the associated weightingfactors:

$\begin{matrix}{I_{nm}^{G} = {\sum\limits_{j}^{\;}{\omega_{j}I_{nm}^{j}}}} & {II}\end{matrix}$S_(nm) is the complex spectrum of the mask, i.e., the Fourier transformof the transmission function of the mask. The transmission function ofthe mask is obtained as a continuous function {circumflex over (M)}(x,y) from the pattern specifications of the mask.

P_(nm) ^((j)) is the complex transmission function of the pupil, whichdescribes the numerical aperture, aberrations and the illuminationangle.

It is not generally possible to calculate the complex spectrum of anarbitrary continuous function {circumflex over (M)}(x, y) analytically.In a variant of the method, the continuous transmission function of themask is discretized. It is thereby transformed into a matrix {circumflexover (M)}_(nm)={circumflex over (M)}(x_(n), y_(n)) composed of n×mpixels. The complex spectrum of this transmission function can becalculated by:

$\begin{matrix}{S_{nm} = {\sum\limits_{n^{\prime} = 1}^{N}{\sum\limits_{m^{\prime} = 1}^{M}{{\hat{M}}_{nm}{\mathbb{e}}^{{- 2}\pi\;{{\mathbb{i}}{({n^{\prime} - 1})}}{{({n - 1})}/N}}{\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({m^{\prime} - 1})}}{{({m - 1})}/M}}}}}} & {III}\end{matrix}$

To permit subpixel-accurate analysis of the simulated aerial image, theintensities of the individual pixels of the simulated aerial image arereproduced with an accuracy of 1 nm or better, based on the dimensionsof the mask. The size of a pixel in an aerial image captured with aposition measuring device has an edge length of 25 nm, referred to themask dimensions. The image is composed of a matrix of 1000×1000 pixels;the image field has an edge length of 25 μm.

Hence, the starting point for the simulation of an aerial image is atransmission function of the mask that has a resolution of 1 nm or even0.5 nm per pixel of the captured aerial image. To keep the computingcomplexity low, this resolution is used only for regions that arerelevant for position determination, such as selected regions offeatures, for example.

In a variant of the method, the continuous transmission function of themask {circumflex over (M)}(x, y) is decomposed into basic buildingblocks (pattern), which are described by functions {circumflex over(P)}_(k) (x, y). Fourier transformation of the functions in these formscan then be performed analytically.

The decomposition into building blocks is effected in such a way thateach building block has a constant complex transmission t_(k). Forexample, in the case of a mask having patterns formed by chromium onquartz glass, the chromium-coated regions are decomposed into buildingblocks of a first complex transmission and the chromium-free buildingblocks into regions having a second complex transmission. The continuoustransformation functions of the masks can be expressed as:

$\begin{matrix}{{{{\hat{M}\left( {x,y} \right)} = {\sum\limits_{k}{t_{k}{{\hat{P}}_{k}\left( {x,y} \right)}}}},{where}}{{{\hat{P}}_{k}\left( {x,y} \right)} = \left\{ \begin{matrix}{1:{{if}\mspace{14mu}\left( {x,y} \right)\mspace{14mu}{is}\mspace{14mu}{within}\mspace{14mu} a\mspace{14mu}{building}\mspace{14mu}{block}}} \\{0:{otherwise}}\end{matrix} \right.}} & {IV}\end{matrix}$

Suitable building blocks are triangles. The mask patterns of currentmasks consist of polygons that can be decomposed into triangles.Analytical Fourier transformation of triangular areas is also possible.The complex spectrum S_(nm) can be calculated as follows:

$\begin{matrix}\begin{matrix}{S_{nm} = {{\frac{1}{ab}{\int_{0}^{a}{\int_{0}^{b}{{\hat{M}\left( {x,y} \right)}{\mathbb{e}}^{{- 2}\pi\;{{\mathbb{i}}{({{{nx}/a} + {{my}/b}})}}}{\mathbb{d}x}{\mathbb{d}y}}}}} =}} \\{\frac{1}{ab}{\int_{0}^{a}{\int_{0}^{b}{\left( {\sum\limits_{k}^{\;}{t_{k}{{\hat{P}}_{k}\left( {x,y} \right)}}} \right){\mathbb{e}}^{{- 2}\pi\;{{\mathbb{i}}{({{{nx}/a} + {{my}/b}})}}}{\mathbb{d}x}{\mathbb{d}y}}}}} \\{{= {\sum\limits_{k}^{\;}{t_{k}S_{nm}^{k,{Triangle}}}}},}\end{matrix} & V\end{matrix}$where a and b denote the edge lengths of the rectangular area to besimulated. This method eliminates the need to discretize the continuoustransmission function of the mask. At the same time, the discretizedspectrum S_(nm) is calculated in the Fourier space.

To decompose the transmission function of the mask into triangles, aknown algorithm is used. See H. ElGindy, H. Everett and G. T. Toussaint,“Slicing an ear using prune-and-search,” Pattern Recognition Letters14/9 (1993), 719-722. This can also be done manually. An example of thedecomposition of a transmission function of a mask 15 into triangularbuilding blocks is illustrated in FIG. 21. This shows only thetriangulation of a region 16 of complex transmission, represented by thehatched areas in FIG. 21. The figure shows the decomposition intotriangles of a cross-shaped marker 16 with a square opening at itscenter. The decomposition of region 17 of the mask 15, depicted in whiteand having a different complex transmission, is not illustrated in FIG.21.

The distortions due to the objective lens of the position measuringdevice are determined by known methods and are present, for example, asa Zernike polynomial. The intensity distribution of the field plane canbe measured with the position measuring device itself by taking anaerial image without a mask in the beam path. Alternatively, a mask withno patterns applied to it is brought into the beam path. To measure theintensity distribution of the pupil plane of the position measuringdevice 10, the Bertrand lens 13 is brought into the beam path. Thethree-dimensional topography of the mask is determined, for example, bymethods such as atomic force microscopy. The complex refractive index ofthe mask material is determined, for example, by ellipsometry.

In a variant of the method, the position of a feature in the first(captured) aerial image or the second (simulated) aerial image isdetermined by determining the positions of selected edges of thefeature. The edge positions are determined by a threshold value methodthat permits subpixel-accurate position determination.

The analysis will be explained based on the example of an edge of anaerial image of a feature configured as a cross composed of two bars L1and L2. The aerial image is depicted in FIG. 2; the darker the shade ofthe pixels in question, the higher their intensity.

The discrete intensity profile along section line A-A in FIG. 2 isillustrated schematically in FIG. 3, a circle being drawn in for eachintensity value of a pixel. For easier comprehensibility, however, notall the pixels are shown. The coordinate in μm is plotted on the x-axisin FIG. 3, and the intensity I in relative units along the y-axis. FIG.3 also shows the threshold value T for the intensity, which is set inorder to determine the position of the edge. In the example describedhere, the width of the region captured per pixel in the x-direction is25 nm, and the edge is therefore captured with this resolution of 25 nmin the x-direction. Each pixel P is assigned the center as itsx-direction coordinate.

As is evident from the representation of FIG. 3, both the left and theright edges lie between two pixels. Region C is shown enlarged in thenext figure, FIG. 4, where the pixels of the discrete intensity profileare numbered consecutively from 1 to 9.

First, the pixel P whose intensity value is closest to the thresholdvalue T is selected. In the example of FIG. 4, this is the fourth pixelP(4). This fourth pixel P(4) and the two adjacent pixels on each side,P(2), P(3), P(5) and P(6), are selected as the analysis pixels.

A line of best fit g1 is calculated for these five analysis pixelsP(2)-P(6), as illustrated in FIG. 5. The intersection point S1 of theline of best fit g1 with the threshold value T or threshold value line Tis then determined, as is the distance δx1 in the x-direction betweenthe intersection point S1 and the pixel P(4) whose intensity value isclosest to the threshold value T.

It is then determined whether the magnitude of the distance δx1 isgreater than a maximum value e.

If this is not the case, the determined distance δx1 and any otherdistances that have been determined, as will be described below, arethen added to the coordinate of pixel P(4) to obtain the edge positionwith subpixel accuracy.

If δx is greater than e, a continuous profile function p(x) of the edgeis shifted by −δx1. The continuous profile function p(x) of the edge,which is calculated as a function of the coordinate based on pixelsP(1)-P(9), is plotted in FIG. 6. Here, the profile function p(x) isobtained, for example, by linear interpolation between two adjacentpixels P on each side. The shift of the continuous profile function p(x)is illustrated in FIG. 7, the shifted profile function p′(x) beingrepresented in dotted form. The intensity value of the shifted profilefunction p′(x) is now determined at the respective coordinates of thepixels P, such that all the pixels P(1)-P(9) have modified intensityvalues. The new intensity values are represented in dotted form.

The pixel whose intensity value is closest to the threshold value T isthen re-determined. This is again pixel P(4). Pixel P(4) and the twoadjacent pixels in both directions are selected as analysis pixels. Aline of best fit g2 is again determined for these analysis pixelsP(2)-P(6), as illustrated in FIG. 8. The intersection point S2 of thelines of best fit g2 with the threshold value is determined, as is thedistance δx2 of the intersection point S2 from the coordinate of themiddle pixel P(4). If the determined distance δx2 is not greater thanthe maximum value e, the process is terminated and the edge position isdetermined with subpixel accuracy based on all the determined distancesδx, as described above.

If, however, the determined distance δx2 is greater than the maximumvalue e, the already shifted profile function is shifted by −δx₂, thenew intensity values of the pixels P are determined in the same manneras in FIG. 7, and the distance δx is then determined in the same manneras in FIG. 8. This is repeated until δx is no longer greater than themaximum value e. The position of the edge is then determined withsubpixel accuracy from all the determined distances.

The position of the right edge can also be determined with subpixelaccuracy by the above procedure. In particular, it is also possible toselect the arithmetic mean of the two edge positions as the position ofthe perpendicular bar L1 (FIG. 2) of the feature.

The positions of the edges can also be determined as follows. Proceedingfrom the discrete intensity profile of FIG. 2, the continuous profilefunction p(x) of the edge is calculated for region C as a function ofthe coordinate x, and thus based on pixels P(1)-P(9), as illustrated inFIG. 9.

The intersection point S3 of the continuous profile function p(x) withthe threshold value T is then determined. The coordinate x3 of theintersection point S3 serves as the starting point for the determinationof additional analysis pixels whose distance from coordinate x3 in eachcase is a whole multiple of the spacing of the pixels P. Here, twoadditional pixels to the right and to the left are determined, asindicated in FIG. 10. The intensity values of the profile function p(x)at the thus-selected coordinates are determined, and five analysispixels A(1)-A(5) are designated in this manner.

A line of best fit g3 for the analysis pixels A(1)-A(5) is determined,as is the intersection point S4 of the lines of best fit g3 with thethreshold value T (FIG. 11). The coordinate x(S4) of the intersectionpoint S4 is then used to calculate the edge position with subpixelaccuracy.

The edge position can further be determined as follows. First, from thediscrete intensity of profile of FIG. 2, region C is selected (FIG. 12).The so-selected pixels P form a discrete pixel profile, which issmoothed by calculating a sliding average over, in this case, fivepixels. The thus smoothed profile is illustrated in FIG. 13, the newintensity values of the individual pixels being represented as dottedlines.

Based on the smoothed pixels P, a continuous profile function p(x) iscalculated as a function of the coordinate x. The intersection point ofthe continuous profile function p(x) with the threshold value T isdetermined and the corresponding coordinate x(S5) of the intersectionpoint S5 is then used to perform the subpixel-accurate positiondetermination, as illustrated in FIG. 14.

Alternatively, the method disclosed in DE10047211 can be used for theedge position determination.

Various methods are available for determining the position of a regionof an aerial image relative to another aerial image. The region can becut out of the aerial image and the position determined in the otheraerial image. As explained subsequently below, it may be advantageousnot to cut out the region, but to mask it in the original aerial image.Its size is preserved in this way, and the position determination methodis simplified if the aerial images to be compared are the same size. Theregion will be referred to below as image 1, and the aerial image asimage 2.

In a first method, particularly when the existing image 1 and image 2are the same size, cross-correlation is used, as disclosed in DE 10 2006059 431 A1.

To determine the relative position of image 1 with respect to image 2,the two-dimensional correlation between image 1 and image 2 iscalculated according to the following formula (1):

$\begin{matrix}\begin{matrix}{C_{m,n} = {{Re}\left( {\frac{1}{P \cdot Q}{\sum\limits_{p = 1}^{P}{\sum\limits_{q = 1}^{Q}{{\hat{a}}_{p,q} \cdot b_{p,q}^{*} \cdot \left( \mu_{p} \right)^{1 - m} \cdot \left( v_{q} \right)^{1 - n}}}}} \right)}} \\{= {\sum\limits_{m^{\prime} = 1}^{P}{\sum\limits_{n^{\prime} = 1}^{Q}{{\hat{A}}_{m^{\prime},n^{\prime}} \cdot B_{{m^{\prime} + 1 - m},{n^{\prime} + 1 - n}}}}}}\end{matrix} & (1)\end{matrix}$Where, in the ideal case, B_(m,n)∝Â_(m,n) and the maximum occurs atm=1=n with the shifted image according to:

$\begin{matrix}{{\hat{A}}_{m,n} \equiv {{{Re}\left( {\sum\limits_{m = 1}^{P}{\sum\limits_{n = 1}^{Q}{{\hat{a}}_{p,q} \cdot \left( \mu_{p} \right)^{1 - m} \cdot \left( v_{q} \right)^{1 - n}}}} \right)}.}} & (2)\end{matrix}$Image 1 and image 2 are the same size and have P rows and Q columns ofpixels, the phasors being defined as follows:μ_(p) =e ^(−2πi·η) ^(p) , p=1 . . . P  (3)ν_(q) =e ^(−2πi·ξ) ^(q) , q=1 . . . Q  (4)with the following coordinates

$\begin{matrix}{\eta_{p} = {\frac{p - 1}{P} - {{floor}\left( {\frac{p - 1}{P} + \frac{1}{2}} \right)}}} & (5) \\{\xi_{q} = {\frac{q - 1}{Q} - {{{floor}\left( {\frac{q - 1}{Q} + \frac{1}{2}} \right)}.}}} & (6)\end{matrix}$The floor function returns the largest whole number of the argument thatis less than or equal to the argument.

In a next step (Step 3), the total number of pixels ({circumflex over(m)}≠1, {circumflex over (n)}≠1) for which the correlation is maximal(Ĉ=C_({circumflex over (m)},{circumflex over (n)})) is determined.

In the following step 4, the subpixel shift vector (Δx, Δy) isdetermined. Two alternatives for this step are described below.

According to a first alternative (step 4a), the so-called antisymmetryof the correlation matrix is minimized. For this purpose, with referenceto a center of symmetry ({circumflex over (m)}, {circumflex over (n)})for each pixel (m, n) and the correspondingly mirrored pixel(2·{circumflex over (m)}−m,2·{circumflex over (n)}−n), the squareddifferences(C_(m,n)−C_(2·{circumflex over (m)}−m,2·{circumflex over (n)}−n))² aresummed and are denoted asAoC_({circumflex over (m)},{circumflex over (n)}).

AoC_({circumflex over (m)}±1,{circumflex over (n)}±1) is the calculatedusing adjacent pixels as the center. The two-dimensional subpixel shiftvector (Δx≈{circumflex over (n)}−1, Δy≈{circumflex over (m)}−1) is thedetermined by means of a parabolic fit or a parabolic interpolation nearthe extreme value m={circumflex over (m)}, n={circumflex over (n)}.

According to the second alternative (step 4b) of step 4, the correlationC_(m,n) for the pixels m={circumflex over (m)}, {circumflex over (m)}±1and n={circumflex over (n)}, {circumflex over (n)}±1 are used directlyto determine the subpixel shift vector (Δx, Δy). In this case, themaximum of the correlation is determined by means of a parabolic fitnear the extreme value m={circumflex over (m)}, n={circumflex over (n)}.

The subpixel deviation (Δx, Δy) determined in step 4a or 4b is added tothe pre-existing shift X_(total) or Y_(total) (step 5) in order tocalculate a new total shift X_(total), Y_(total):x _(total) →x _(total) +Δx; y _(total) →y _(total) +Δy  (7)

On the first pass through step 4, the pre-existing total shift is theshift of (m, n) to (1,1). In the subsequent iteration loops, thepre-existing total shift is the total shift calculated in step 5 of thepreceding iteration loop.

From this total shift, the phase tilt of the spectrum is generated bymultiplication (Step 6):â _(p,q) ≡a _(p,q)·(μ_(p))^(−y) ^(total) ·(ν_(q))^(−x) ^(total)   (8)

A check is then performed to determine whether the subpixel shift (Δx,Δy) is less than a predetermined maximum value (step 7). If this is notthe case, the correlation C_(m,n) is calculated according to the aboveformula 1 with the phase tilt according to formula 8 (step 8), which isequivalent to shifting the two images, image 1 and image 2, relative toeach other by the total shift determined in step 5.

Steps 4-7 are then repeated, although it is no longer necessary for theshift to be a full pixel shift (step 3). In the re-performed step 6, thenew total shift is re-applied to image 2. In terms of result, this isthe same as shifting the already shifted image 2 only by therecalculated subpixel shift vector. Formula (7) is to be replaced by thefollowing formula (7′) in this case:Δx _(total) =Δx, y _(total) =Δy  (7′)

Steps 8 and 4-7 are repeated until the subpixel shift in step 7 is foundto be smaller than a defined threshold value.

If this is the case, no further iterations are performed, and, in step9, the total shift or total image distance (x_(total), y_(total))between the two images, image 1 and image 2, is output.

To determine the position of image 1 (a region of an aerial image)relative to image 2 (another aerial image), in a further inventivemethod the method of least squares is used. This mathematical method isadvantageous because, unlike the preceding mathematical method, it canbe employed even when discontinuities are present in the intensitydistribution of the edge region of image 1. As a result, it is possiblenot only to select regions that encompass an entire feature, but also toselect regions that include, for example, an edge of a feature. In image1, the selected region of the aerial image is masked. All the image dataoutside the masked region are suppressed during the positiondetermination. Image 1 and image 2 thus remain the same size. Pluralregions, for example plural edges of a feature, can also be masked inimage 1.

The position is determined by the least squares method, image 1 beingshifted cyclically in relation to image 2 and the sum of the squareddifferences between the intensities in the two images, image 1 and image2, being calculated for each shifted position across all the image dots.Summing the squared differences between the intensities in image 1 andimage 2 across all the image dots is equivalent to overlaying the twoimages, image 2 being stationary and image 1 being shifted.

The term “cyclic shift” is understood here to mean that the image datafrom the first image, image 1, which during the shift are shiftedoutside the image edge of the second image, image 2, are shifted backinto the respective opposite image edge side of image 2.

The shifted position is sought at which the above-described weightedsum, which will be referred to below as the optimization function,assumes an extreme value (maximum or minimum). The shifted positionpresent at this extreme value is the sought position of image 1 relativeto image 2.

An optimization function M(x,y) can be represented, for example, by thefollowing formula (1a):

$\begin{matrix}{{M\left( {x,y} \right)} = {- {\sum\limits_{m = 0}^{P - 1}{\sum\limits_{n = 0}^{Q - 1}{K_{m,n} \cdot {\left\lbrack {{A_{m,n}\left( {x,y} \right)} - B_{m,n}} \right\rbrack^{2}\overset{({x,y})}{\longrightarrow}{maximum}}}}}}} & \left( {1a} \right)\end{matrix}$where K_(m,n), the normalized 2D masking (also referred to hereinafteras keyhole masking or keyhole apodization), with

${\sum\limits_{m = 0}^{P - 1}{\sum\limits_{n = 0}^{Q - 1}K_{m,n}}} = 1.$A_(m,n)(x,y) represents the image 1 to be shifted, which has beenshifted by a vector (−x,−y), and B_(m,n) denotes the stationary image 2.Both captured images are the same size and each contains P×Q pixels.Both captured images are matrices composed of P×Q pixels of the samesize. In a variant of the exemplary embodiment, P=Q=1000. Thiscorresponds to the matrix of pixels of the detector 6 of the positionmeasuring device 10. The intensity values of the two captured imagesthat can be found at the same pixel position in both captured images(taking the shift (−x,y) into account) are constantly subtracted fromeach another by the summation.

Since B_(m,n) is not shifted and therefore does not depend on the shiftvector (−x,−y), the optimization function can be rewritten as follows:

$\begin{matrix}{{M\left( {x,y} \right)} = {\sum\limits_{m = 0}^{P - 1}{\sum\limits_{n = 0}^{Q - 1}{K_{m,n} \cdot {\left\lbrack {{2 \cdot {A_{m,n}\left( {x,y} \right)} \cdot B_{m,n}} - {A_{m,n}\left( {x,y} \right)}^{2}} \right\rbrack\overset{({x,y})}{\longrightarrow}{maximum}}}}}} & \left( {2a} \right)\end{matrix}$

If a spectral decomposition is performed for image 1 and image 2,formula 2a can be written as follows:

$\begin{matrix}{{M\left( {x,y} \right)} = {\frac{1}{P \cdot Q}{\sum\limits_{p,q}{\left\lbrack \overset{\overset{= {Kernel}_{p,g}}{︷}}{{2 \cdot a_{p,q} \cdot \beta_{p,q}^{*}} - \alpha_{p,q}} \right\rbrack \cdot {{\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({{x \cdot \xi_{q}} + {y \cdot \eta_{p}}})}}}\overset{({x,y})}{\longrightarrow}{maximum}}}}}} & \left( {3a} \right)\end{matrix}$where the asterisk (*) here means that the quantity concerned is acomplex conjugate value.

For the above Formula 3a, the following spectral decompositions areperformed:

$\begin{matrix}{{A_{m,n}\left( {x,y} \right)} = {\frac{1}{P \cdot Q}{\sum\limits_{p,q}{a_{p,q} \cdot {\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({{{\lbrack{n + x}\rbrack} \cdot \xi_{q}} + {{\lbrack{m + y}\rbrack} \cdot \eta_{p}}})}}}}}}} & \left( {4a} \right) \\{{A_{m,n}\left( {x,y} \right)}^{2} = {\frac{1}{P \cdot Q}{\sum\limits_{p,q}{\alpha_{p,q} \cdot {\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({{{\lbrack{n + x}\rbrack} \cdot \xi_{q}} + {{\lbrack{m + y}\rbrack} \cdot \eta_{p}}})}}}}}}} & \left( {5a} \right) \\{{K_{m,n} \cdot B_{m,n}} = {\frac{1}{P \cdot Q}{\sum\limits_{p,q}{\beta_{p,q} \cdot {\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({{n \cdot \xi_{q}} + {m \cdot \eta_{p}}})}}}}}}} & \left( {6a} \right) \\{K_{m,n} = {\frac{1}{P \cdot Q}{\sum\limits_{p,q}{\kappa_{p,q} \cdot {\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({{n \cdot \xi_{q}} + {m \cdot \eta_{p}}})}}}}}}} & \left( {7a} \right)\end{matrix}$In addition, a_(p,q) denotes the following spectral folding:

$\begin{matrix}{{\alpha_{p,q} \equiv \left( {a \otimes a} \right)_{p \cdot q}} = {\frac{1}{P \cdot Q}{\sum\limits_{p^{\prime},q^{\prime}}{a_{{p - p^{\prime}},{q - q^{\prime}}} \cdot a_{p^{\prime},q^{\prime}}}}}} & \left( {8a} \right)\end{matrix}$and the functions ξ_(q), η_(p) are defined according to the followingformulas 9a and 10a:

$\begin{matrix}{{\xi_{q} = {\frac{q}{Q} - {{floor}\left( {\frac{q}{Q} + \frac{1}{2}} \right)}}};{q = {{0\mspace{14mu}\ldots\mspace{14mu} Q} - 1}}} & \left( {9a} \right) \\{{\eta_{p} = {\frac{p}{P} - {{floor}\left( {\frac{p}{P} + \frac{1}{2}} \right)}}};{p = {{0\mspace{14mu}\ldots\mspace{14mu} P} - 1}}} & \left( {10a} \right)\end{matrix}$The floor function returns the largest whole number of the argument thatis smaller than or equal to the argument. In this way, the relativeposition of image 1 with respect to image 2 can be determinednumerically with high accuracy.

Due to the weighted summation according to the above-cited optimizationfunction, a hard edge is incorporated into the overlay, but this doesnot eliminate bandwidth limitation from the spectral decomposition.Thus, in the difference according to the kernel_(p,q) in formula 3a,even though neither β*_(p,q) nor κ_(p,q) is bandwidth-limited, both theminuend and the subtrahend are bandwidth-limited, since a_(p,q) andα_(p,q) are spectral bandwidth-limited. Due to the folding according toformula 8, α_(p,q) does have a larger spectral region than a_(p,q) butit is still spectral band-limited. It is therefore possible to obtain anumerical solution to formula 3a that is very accurate. For the rest,the subtrahend α·κ* can be viewed as a term that balances the effect ofthe leading cross-correlation term 2a·β*. Hence, intensityinhomogeneities can be present on the edge trajectory of the weightingand the inventive method will still yield the relative position beingsought. In conventional correlation methods, for example of the kinddescribed in DE 10 2007 033 815 A1, edge inhomogeneities of this kindmake it impossible to obtain meaningful results.

Not all of the shifted positions have been found to yield meaningfulresults, since the optimization function often has several local maximaor minima. Consequently, for example the relative position of image 1with respect to image 2 or a portion thereof is determined relativelycoarsely by correlation methods, as described above. Such relativelycoarse position determination is understood here particularly to includepixel-accurate determination. This relative position shift is then usedas the starting value for the optimization function, to ensure that theextreme value that is then found is also the sought local extreme valueof the optimization function.

Other parameters can also be taken into account in this least squaresmethod with weighting. For example, the disparate image control of thetwo images can be taken into account by means of a further parameter S,as given in the following formula 11a. This is necessary, for example,if the intensity values of the images to be compared, image 1 and image2, are not normalized to a reference intensity. Further additionally, abrightness offset between the two images, due to stray light, forexample, can also be taken into account, as expressed by parameter T informula 12a. These formulas 11a and 12a can be represented spectrally,as described earlier hereinabove, and can thus be well solvednumerically.

$\begin{matrix}{{M\left( {x,y,S} \right)} = {- {\sum\limits_{m = 0}^{P - 1}{\sum\limits_{n = 0}^{Q - 1}{K_{m,n} \cdot {\left\lbrack {{A_{m,n}\left( {x,y} \right)} - {S \cdot B_{m,n}}} \right\rbrack^{2}\overset{{({x,y})},S}{\longrightarrow}{maximum}}}}}}} & \left( {11a} \right) \\{{M\left( {x,y,S,T} \right)} = {- {\sum\limits_{m = 0}^{P - 1}{\sum\limits_{n = 0}^{Q - 1}{K_{m,n} \cdot {\left\lbrack {{A_{m,n}\left( {x,y} \right)} - {S \cdot B_{m,n}} - T} \right\rbrack^{2}\overset{{({x,y})},S,T}{\longrightarrow}{maximum}}}}}}} & \left( {12a} \right)\end{matrix}$

The optimization function M according to formulas 11a and 12a can, inturn (as in formula 3a), be represented as follows:

$\begin{matrix}{{{M\left( {x,y} \right)} = {\frac{1}{P \cdot Q}{\sum\limits_{p,q}{{Kernel}_{p,q} \cdot {\mathbb{e}}^{2\pi\;{{\mathbb{i}}{({{x \cdot \xi_{q}} + {y \cdot \eta_{p}}})}}}}}}},} & \left( {13a} \right)\end{matrix}$only the kernel_(p,q) being different. For formula 11a, the kernel_(p,q)is stated as follows:Kernel_(p,q)=−α_(p,q)·κ*_(p,q)+((a·{circumflex over (β)}*)

(a·{circumflex over (β)}*))_(p,q)  (14a),the following abbreviated notation being used here in addition to theabbreviations stated above:

$\begin{matrix}{{\hat{B}}_{m,n} \equiv {B_{m,n}/\sqrt{\sum\limits_{m^{\prime},n^{\prime}}{K_{m^{\prime},n^{\prime}} \cdot B_{m^{\prime},n^{\prime}}^{2}}}}} & \left( {15a} \right) \\{{\hat{\beta}}_{p,q} \equiv {\sum\limits_{m,n}{K_{m,n} \cdot {\hat{\beta}}_{m,n} \cdot {\mathbb{e}}^{{- 2}\pi\;{{\mathbb{i}}{({{n \cdot \xi_{q}} + {m \cdot \eta_{p}}})}}}}}} & \left( {16a} \right)\end{matrix}$The kernel_(p,q) for formula 12a is thus represented as follows:Kernel_(p,q)=−α_(p,q)·κ*_(p,q)+((a·{tilde over (β)}*)

(a·{tilde over (β)}*))_(p,q)+((a·κ*)

(a·κ*))_(p,q)   (17a)the following abbreviated notation being used here in addition to theabbreviations stated above:

$\begin{matrix}{{\overset{\sim}{B}}_{m,n} \equiv \frac{B_{m,n} - \left( {\sum\limits_{m^{\prime},n^{\prime}}{K_{m^{\prime},n^{\prime}} \cdot B_{m^{\prime},n^{\prime}}}} \right)}{\sqrt{\left( {\sum\limits_{m^{\prime},n^{\prime}}{K_{m^{\prime},n^{\prime}} \cdot B_{m^{\prime},n^{\prime}}^{2}}} \right) - \left( {\sum\limits_{m^{\prime},n^{\prime}}{K_{m^{\prime},n^{\prime}} \cdot B_{m^{\prime},n^{\prime}}}} \right)^{2}}}} & \left( {18a} \right) \\{{\overset{\sim}{\beta}}_{p,q} \equiv {\sum\limits_{m,n}{K_{m,n} \cdot {\overset{\sim}{B}}_{m,n} \cdot {\mathbb{e}}^{{- 2}{{\pi\mathbb{i}}{({{n \cdot \xi_{q}} + {m \cdot \eta_{p}}})}}}}}} & \left( {19a} \right)\end{matrix}$

The kernel_(p,q) in formula 14a for M(x,y,S) according to formula 11acan be determined by setting the partial derivative of the optimizationfunction M(x,y,S) after S to zero, solving this equation after S andinserting it into formula 11a, such that the optimization for parameterS is already included in the kernel statement.

In the same way, the kernel_(p,q) in formula 17a can be determined forformula 12a by deriving the optimization function M(x,y,S,T) oncepartially to S according to formula 12a and setting the derivative tozero and deriving it once partially to T and setting the derivative tozero, and inserting the thus-determined equations for S and T intoformula 12a.

A flow chart of the method according to the invention for determiningthe registration error is illustrated in FIG. 20. To determine aregistration error of a feature, a first aerial image that includes afeature is taken at a defined position P1 on the mask. Position P1 givesthe location of the origin of the image field of the position measuringdevice 10 in the mask coordinate system, which, as noted earlier, is atthe center of the image field. Thus, the origin of the first aerialimage is likewise located at the center thereof. The dimensions of theaerial image correspond to the image field on the mask.

The simulation of the second aerial image is based on the patternspecifications of the mask (the mask design) at position P1,corresponding to the dimensions of the image field or first aerialimage.

In a variant of the method, the position of the feature is defined bythe positions of characteristic edges of the feature. The characteristicedges are defined as regions in the first (or second) aerial image. Thisis performed by the user at a video monitor (not shown in FIG. 1) whichis part of the control unit 7 and which displays the aerial images. Anexample will be provided farther below with reference to FIG. 18a . Thepositions of the defined regions are given in the coordinate system ofthe aerial images. The first positions of the edges, determinedaccording to the aforementioned threshold value method, are also givenin the coordinate system of the aerial image. Since the registrationerrors to be expected are small, the same regions are used for positiondetermination in the second (first) aerial image as in the first(second) aerial image.

Based on the distances of the second positions of the edges in thesecond (first) aerial image from the first positions of the edges in thefirst (second) aerial image, the registration errors of the edges aredetermined in the coordinate system of the aerial images, but can beconverted directly to the mask coordinate system. If the registrationerrors of plural edges are determined, these are used to determine aresulting registration error of the feature.

To determine a registration error of a feature from plural edgepositions, in a variant of the method, from the positions of the edgesin an aerial image, a characteristic point of the feature, such as itscenter of symmetry, for example, is selected as the position of thefeature. A method for determining the center of symmetry of a crosscomposed of bars is given farther below, and this method can also beused generally for suitable features. The first position of the featureis determined in the first aerial image and the second position of thefeature is determined in the second aerial image. The registration erroris then calculated as the difference between the coordinates of thesecond position and the coordinates of the first position.

The aforesaid correlation method can also be used on the basis of theedges defined in the first (second) aerial image. In this case, theposition of the region of an edge in the second (first) aerial image isdetermined. The positions are then compared by means of defined points,such as, for example, the centers of the regions. Based on the distancesof the centers of the regions in the first aerial image from the centersof the regions in the second aerial image, the registration error of anedge is determined initially in the coordinate system of the aerialimages, but can be converted directly to the mask coordinate system.From the registration errors of the edges, the registration error of thefeatures can be determined. To accomplish this, the registration errorsof the edges and the positions of the edges as specified in the designare used to calculate the second positions of the edges (the actualpositions). From these second positions, as explained above, the firstposition of the feature is then determined. The first position of thefeature is determined analogously from the edge positions given in thepattern specifications.

In a variant of the method, plural regions of edges of a feature cansimultaneously be correlated with the second aerial image. For thispurpose, as described above, the regions of the edges of interest aremasked in the second aerial image.

In a variant of the method, in the first (second) aerial image, theregion of an entire feature whose registration error is to be determinedis defined. The position of the region in the second (first) aerialimage is then determined. The positions are then compared on the basisof a defined point in the regions, such as their centers, for example.Based on the deviation of the coordinates of the center of the region inthe first aerial image from the coordinates of the center of the regionin the second aerial image, the registration error is determinedinitially in the coordinate system of the aerial images, but can beconverted directly to the mask coordinate system.

The determination of registration errors will be explained withreference to a first example of a cross with an asymmetricalenvironment. The value to be determined is the registration error of anintersection point 31 of two bars forming a cross 30. The mask patternis illustrated in FIG. 15a ; hatched regions 26, 27, 28, 29 arenon-transmitting or non-reflecting regions of the mask. In aconventional transmissive mask, the hatched regions are coated withchromium. A nearly white cross 30 composed of two bars of the samewidth, which form four sections, is adjacent, by four edges of the bars,to two black squares. The other four edges of the bars are adjacent tofour nearly black bars 28, 29 that are adjacent to white squares 36, 37.The resulting first or second aerial image 25′ is illustrated in FIG.15b . The lighter the shade of gray, the higher the intensity of theparticular region or pixel.

To determine the position of the intersection point 31 of the cross 30,first, the positions of the eight long edges of the four sections of thecross 30 are determined. Eight regions of interest (ROIs) 32 to 35 aredefined for this purpose by the user, as illustrated in FIG. 2. Theregions of interest are defined in either the first or the second aerialimage. The edge position determination is performed on both the firstand the second aerial image. The two regions 33, 35 in the x-directionserve to determine the position of the bar in the y-direction. The tworegions 32, 34 in the y-direction serve to determine the position of thebar in the x-direction. For the regions in the x-direction, an edgeprofile in the y-direction is determined by calculating the average ofall the intensity values in the x-direction. Each edge profile receivesthe profiles of two opposite edges of a section of the bars, which areselected during the determination. From the edge profiles, the positionsof the edges in the y-direction are determined with subpixel accuracyaccording to the above-described threshold value method. The thresholdvalue is, for example, 70% of the maximum intensity in the aerial image.An analogous procedure is used to determine the positions of the edgesin the x-direction. From the positions of the eight edges, the center 31of the cross 30 can now be determined by statistical or geometricalcalculation. The distance between the two center points 31 determined inthe first and second aerial images 25 is the registration error.

The following method can be used to determine the position of the center31 of the cross 30. The directions of extent of the edges are taken fromthe pattern specifications of the mask. The orientation of the edges isascertained by means of the determined edge positions. The tangentequations of the edges are therefore known. All the edges so definedform a region around the center 31 of the cross 30. The center is nowcalculated by the least squares method in such a way that its shortestdistances from all the tangents become minimal in the sum of thesquares.

To clarify the falsification of positions by proximity effects, aerialimages of various embodiments of the mask 25 were simulated. The width39 of each non-transmissive region 28, 29 of the mask located adjacent arespective square transmissive region 36, 37 was varied. The width 39 ofthe perpendicularly disposed leg of regions 28, 29 is constant. In thedrawing, this is defined as 200 nm. The line width of the cross 38 onthe mask has a constant value of 165 nm. The distance of the squareregions 36, 37 from the cross 30 is therefore varied. The registrationerror of the center of the cross 30 that is introduced as an artifact bythe proximity effects in the y-direction is negligible, since theenvironment is symmetrical in this direction. In the x-direction,however, the proximity effects are considerable. FIG. 16 illustrates thefalse registration error as a function of the distance, i.e., the widthof the leg of the non-transmissive regions 28, 29. With a distance of100 nm, the false registration error is 6.5 nm; with a distance of, forexample, 400 nm, it is 0.8 nm. The false registration error tends towardzero with increasing distance between the transmissive regions.

In a second example, the centers of symmetry of pinholes are determined.The mask pattern specifications include five square pinholes arranged ina straight line in the x-direction. The mask pattern is illustrated inFIG. 17. The edge length of the pinholes is 120 nm; the distance betweenpinholes is 240 nm. The first and second aerial images of the mask areshown in FIGS. 18a and 18b , respectively.

The determination of the positions of the centers of symmetry of thepinholes 46 to 50 is illustrated taking the example of the centralpinhole 48 in FIGS. 18a and 18b . The positions of the four edges of thepinhole can be determined. For this purpose, four regions 51 to 54 ofthe particular edges are selected from the aerial image 45′, asillustrated in FIG. 18a , and the positions are determined by thethreshold value method as explained above. The lighter the shade of graydepicted in the aerial image, the higher the intensity of the particularregion or pixel. From the positions of the four edges, the position ofthe center of symmetry of the pinhole is then calculated using simplegeometrical calculations. To do this, for the x-position of the centerof symmetry, the average of the two determined x-values of the edgesextending in the y-direction is calculated. For the y-position of thecenter of symmetry, the average of the two determined y-values of theedges extending in the x-direction is calculated. The position of thecenter of symmetry of the pinhole is determined by the same method forthe other aerial image. The difference between the respective locationsof the centers of symmetry is the registration error.

The differences between the positions of the edges can also bedetermined by the above-cited correlation method. The registration errorcan also be calculated directly from the differences between the edgepositions. For this purpose, the average registration error iscalculated in the x-direction and the y-direction, respectively.

It is also possible, however, to use the edge positions from the maskpattern specifications to calculate the nominal position of the centerof symmetry of the pinhole, as explained above. The actual positions ofthe edges are calculated from the nominal positions of the edges and theregistration errors of the edge positions. From the nominal positions ofthe edges, the nominal position of the center of symmetry of the pinholeis then calculated as described. The registration error is thencalculated, as described above, as the difference between the positionsof the centers of symmetry.

The position of a center of symmetry of a pinhole is also determined bydetermining the position of the pinhole from one aerial image in theother aerial image. To do this, region 55 of the pinhole is selected inone aerial image, as illustrated in FIG. 18b . The position of thisregion in the other aerial image is then determined as explained above.The difference between the positions corresponds to the registrationerror.

The influence of proximity effects on registration error measurement hasbeen investigated with regard to this example, as well. It was foundthat registration errors in the x-direction introduced as artifacts byproximity effects are more pronounced in the case of pinholes having anasymmetrical proximity environment. FIG. 19a is a graph of the falseregistration errors for each of the five pinholes 46 to 50. These havebeen numbered (in the same order) 1 to 5 in the x-direction. For themiddle pinhole 48 (number 3), the false registration error is nearlyzero; it increases to 3.2 nm for the outermost pinholes 46 and 50(numbers 1 and 5). The proximity effects in the y-direction arenegligible, as can be seen from the graph of FIG. 19 b.

The features described above related to processing of data can beimplemented in digital electronic circuitry, or in computer hardware,firmware, software, or in combinations of them. The features can beimplemented in a computer program product tangibly embodied in aninformation carrier, e.g., in a machine-readable storage device, forexecution by a programmable processor; and method steps can be performedby a programmable processor executing a program of instructions toperform functions of the described implementations by operating on inputdata and generating output. Alternatively or addition, the programinstructions can be encoded on a propagated signal that is anartificially generated signal, e.g., a machine-generated electrical,optical, or electromagnetic signal, that is generated to encodeinformation for transmission to suitable receiver apparatus forexecution by a programmable processor.

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made. For example,elements of one or more implementations may be combined, deleted,modified, or supplemented to form further implementations. As yetanother example, the logic flows depicted in the figures do not requirethe particular order shown, or sequential order, to achieve desirableresults. In addition, other steps may be provided, or steps may beeliminated, from the described flows, and other components may be addedto, or removed from, the described systems. Accordingly, otherimplementations are within the scope of the following claims.

What is claimed is:
 1. A method for determining a registration error ofa feature on a mask, the registration error indicating a deviation of anominal position of the feature from its actual position on the mask themethod comprising: providing a first aerial image that was captured by aposition measuring device and includes at least said feature, saidfeature comprising a registration pattern or marker, simulating, frompattern specifications of the mask, a second aerial image that includesat least said feature, taking into account at least one effect thatcauses distortion of said first aerial image, and determining theregistration error of said feature as the distance of the position ofsaid feature in said first aerial image from the position of saidfeature in said second aerial image; wherein said second aerial image issimulated, on the basis of the pattern specifications of the mask, as atransmission function that has, at least in part, a higher resolutionthan said first aerial image.
 2. The method according to claim 1,wherein at least one of the following effects is taken into account inthe simulation of said second aerial image: proximity effects,distortion due to an objective lens of the position measuring device,topography of the mask, and refractive index of the mask.
 3. The methodas in claim 1, wherein the simulation of said second aerial image isbased on pattern specifications of the mask that are in the form of acontinuous transmission function, said second aerial image beingdetermined as a matrix of discrete intensity values, a discretization ofthe spectrum of the mask being performed without further approximation.4. The method as in claim 1, wherein at least one region of said featureis defined that will be taken into account in determining the distance.5. The method as in claim 4, wherein said at least one region of saidfeature is embodied as an edge of said feature.
 6. The method as inclaim 1, comprising: designating, in said first and said second aerialimage, at least one region of said feature that is to be taken intoaccount in determining the registration error, determining the at leastone first and second position of said at least one region, anddetermining the distance on the basis of said at least one first andsecond positions of said at least one region in said aerial images. 7.The method as in claim 6, wherein said at least one region is embodiedas at least one edge of said feature, said at least one position of saidedge being determined by a threshold value method.
 8. The method as inclaim 7, comprising: determining a first position of said feature insaid first aerial image from said at least one first position of anedge, determining a second position of said feature in said secondaerial image from said at least one second position of an edge, anddetermining the distance as the difference between said first positionand said second position.
 9. The method as in claim 1, comprising:designating at least one region in said first or second aerial image,determining a position of said region in the other said aerial image,and determining the distance on the basis of the at least one differencebetween the positions of said at least one region in said aerial images.10. The method as in claim 9, wherein the position of said at least oneregion is determined by the least squares method.
 11. The method as inclaim 9, wherein said at least one designated region is masked, allimage data of the aerial image outside the masked region beingsuppressed.
 12. The method as in claim 9, wherein said one regionincludes said feature.
 13. The method as in claim 1, wherein a firstposition of said at least one edge is adopted as the nominal positionfrom the pattern specifications of the mask, and a second position ofsaid edge is determined as the actual position from the patternspecifications of the mask and from the difference between the positionsof said at least one region in said aerial images.
 14. A positionmeasuring device for determining a registration error of a feature on amask, which device comprises a control unit that performs a method fordetermining a registration error of a feature on a mask, theregistration error indicating a deviation of a nominal position of thefeature from its actual position on the mask, the method comprising:providing a first aerial image that was captured by a position measuringdevice and includes at least said feature, said feature comprising aregistration pattern or marker, simulating, from pattern specificationsof the mask, a second aerial image that includes at least said feature,taking into account at least one effect that causes distortion of saidfirst aerial image, and determining the registration error of saidfeature as the distance of the position of said feature in said firstaerial image from the position of said feature in said second aerialimage, in which the second aerial image is simulated, on the basis ofthe pattern specifications of the mask, as a transmission function thathas, at least in part, a higher resolution than the first aerial image.15. The position measuring device of claim 14 in which at least one ofthe following effects is taken into account in the simulation of thesecond aerial image: proximity effects, distortion due to an objectivelens of the position measuring device, topography of the mask, andrefractive index of the mask.
 16. The position measuring device as inclaim 14 in which the simulation of the second aerial image is based onpattern specifications of the mask that are in the form of a continuoustransmission function, the second aerial image being determined as amatrix of discrete intensity values, a discretization of the spectrum ofthe mask being performed without further approximation.
 17. The positionmeasuring device as in claim 14 in which at least one region of thefeature is defined that will be taken into account in determining thedistance.
 18. The position measuring device as in claim 17 in which theat least one region of the feature is embodied as an edge of thefeature.
 19. The position measuring device as in claim 14 in which themethod performed by the control unit comprises: designating, in thefirst and the second aerial image, at least one region of the featurethat is to be taken into account in determining the registration error,determining the at least one first and second position of the at leastone region, and determining the distance on the basis of the at leastone first and second positions of the at least one region in the aerialimages.
 20. The position measuring device as in claim 19 in which the atleast one region is embodied as at least one edge of the feature, the atleast one position of the edge being determined by a threshold valuemethod.
 21. The position measuring device as in claim 20 in which themethod performed by the control unit comprises: determining a firstposition of the feature in the first aerial image from the at least onefirst position of an edge, determining a second position of the featurein the second aerial image from the at least one second position of anedge, and determining the distance as the difference between the firstposition and the second position.
 22. The position measuring device asin claim 14 in which the method performed by the control unit comprises:designating at least one region in the first or second aerial image,determining a position of the region in the other the aerial image, anddetermining the distance on the basis of the at least one differencebetween the positions of the at least one region in the aerial images.23. The position measuring device as in claim 22 in which the positionof the at least one region is determined by the least squares method.24. The position measuring device as in claim 22 in which the at leastone designated region is masked, all image data of the aerial imageoutside the masked region being suppressed.
 25. The position measuringdevice as in claim 22 in which the one region includes the feature. 26.The position measuring device as in claim 14 in which a first positionof the at least one edge is adopted as the nominal position from thepattern specifications of the mask, and a second position of the edge isdetermined as the actual position from the pattern specifications of themask and from the difference between the positions of the at least oneregion in the aerial images.
 27. A method for determining a registrationerror of a feature on a mask, the registration error indicating adeviation of a nominal position of the feature from its actual positionon the mask, the method comprising: providing a first aerial image thatwas captured by a position measuring device and includes at least saidfeature, said feature comprising a registration pattern or marker,simulating, from pattern specifications of the mask, a second aerialimage that includes at least said feature, taking into account at leastone effect that causes distortion of said first aerial image,designating at least one region in said first or second aerial image,determining a position of said region in the other said aerial image,determining the registration error of said feature as the distance ofthe position of said feature in said first aerial image from theposition of said feature in said second aerial image, and determiningthe distance on the basis of the at least one difference between thepositions of said at least one region in said aerial images, whereinsaid at least one designated region is masked, all image data of theaerial image outside the masked region being suppressed.
 28. The methodof claim 27 in which at least one of the following effects is taken intoaccount in the simulation of said second aerial image: proximityeffects, distortion due to an objective lens of the position measuringdevice, topography of the mask, and refractive index of the mask. 29.The method of claim 27 in which the simulation of said second aerialimage is based on pattern specifications of the mask that are in theform of a continuous transmission function, said second aerial imagebeing determined as a matrix of discrete intensity values, adiscretization of the spectrum of the mask being performed withoutfurther approximation.
 30. The method of claim 27 in which at least oneregion of said feature is defined that will be taken into account indetermining the distance.
 31. The method of claim 30 in which said atleast one region of said feature is embodied as an edge of said feature.32. A method for determining a registration error of a feature on amask, the registration error indicating a deviation of a nominalposition of the feature from its actual position on the mask, the methodcomprising: providing a first aerial image that was captured by aposition measuring device and includes at least said feature, saidfeature comprising a registration pattern or marker, simulating, frompattern specifications of the mask, a second aerial image that includesat least said feature, taking into account at least one effect thatcauses distortion of said first aerial image, and determining theregistration error of said feature as the distance of the position ofsaid feature in said first aerial image from the position of saidfeature in said second aerial image, wherein a first position of said atleast one edge is adopted as the nominal position from the patternspecifications of the mask, and a second position of said edge isdetermined as the actual position from the pattern specifications of themask and from the difference between the positions of said at least oneregion in said aerial images.
 33. The method of claim 32 in which atleast one of the following effects is taken into account in thesimulation of said second aerial image: proximity effects, distortiondue to an objective lens of the position measuring device, topography ofthe mask, and refractive index of the mask.
 34. The method of claim 32in which the simulation of said second aerial image is based on patternspecifications of the mask that are in the form of a continuoustransmission function, said second aerial image being determined as amatrix of discrete intensity values, a discretization of the spectrum ofthe mask being performed without further approximation.
 35. The methodof claim 32 in which at least one region of said feature is defined thatwill be taken into account in determining the distance.
 36. The methodof claim 35 in which said at least one region of said feature isembodied as an edge of said feature.
 37. A position measuring device fordetermining a registration error of a feature on a mask, which devicecomprises a control unit that performs a method for determining aregistration error of a feature on a mask, the registration errorindicating a deviation of a nominal position of the feature from itsactual position on the mask, the method comprising: providing a firstaerial image that was captured by a position measuring device andincludes at least said feature, said feature comprising a registrationpattern or marker, simulating, from pattern specifications of the mask,a second aerial image that includes at least said feature, taking intoaccount at least one effect that causes distortion of said first aerialimage, designating at least one region in the first or second aerialimage, in which the at least one designated region is masked, all imagedata of the aerial image outside the masked region being suppresseddetermining a position of the region in the other the aerial image,determining the registration error of said feature as the distance ofthe position of said feature in said first aerial image from theposition of said feature in said second aerial image, and determiningthe distance on the basis of the at least one difference between thepositions of the at least one region in the aerial images.
 38. Theposition measuring device of claim 37 in which at least one of thefollowing effects is taken into account in the simulation of the secondaerial image: proximity effects, distortion due to an objective lens ofthe position measuring device, topography of the mask, and refractiveindex of the mask.
 39. The position measuring device of claim 37 inwhich the simulation of the second aerial image is based on patternspecifications of the mask that are in the form of a continuoustransmission function, the second aerial image being determined as amatrix of discrete intensity values, a discretization of the spectrum ofthe mask being performed without further approximation.
 40. The positionmeasuring device of claim 37 in which at least one region of the featureis defined that will be taken into account in determining the distance.41. The position measuring device of claim 40 in which the at least oneregion of the feature is embodied as an edge of the feature.
 42. Aposition measuring device for determining a registration error of afeature on a mask, which device comprises a control unit that performs amethod for determining a registration error of a feature on a mask, theregistration error indicating a deviation of a nominal position of thefeature from its actual position on the mask, the method comprising:providing a first aerial image that was captured by a position measuringdevice and includes at least said feature, said feature comprising aregistration pattern or marker, simulating, from pattern specificationsof the mask, a second aerial image that includes at least said feature,taking into account at least one effect that causes distortion of saidfirst aerial image, determining the registration error of said featureas the distance of the position of said feature in said first aerialimage from the position of said feature in said second aerial image,wherein at least one region of the feature is defined that will be takeninto account in determining the distance, and wherein a first positionof the at least one edge is adopted as the nominal position from thepattern specifications of the mask, and a second position of the edge isdetermined as the actual position from the pattern specifications of themask and from the difference between the positions of the at least oneregion in the aerial images.
 43. The position measuring device of claim42 in which at least one of the following effects is taken into accountin the simulation of the second aerial image: proximity effects,distortion due to an objective lens of the position measuring device,topography of the mask, and refractive index of the mask.
 44. Theposition measuring device of claim 42 in which the simulation of thesecond aerial image is based on pattern specifications of the mask thatare in the form of a continuous transmission function, the second aerialimage being determined as a matrix of discrete intensity values, adiscretization of the spectrum of the mask being performed withoutfurther approximation.
 45. The position measuring device of claim 42 inwhich the method performed by the control unit comprises: designating,in the first and the second aerial image, at least one region of thefeature that is to be taken into account in determining the registrationerror, determining the at least one first and second position of the atleast one region, and determining the distance on the basis of the atleast one first and second positions of the at least one region in theaerial images.
 46. The position measuring device of claim 45 in whichthe at least one region is embodied as at least one edge of the feature,the at least one position of the edge being determined by a thresholdvalue method.